Geometric structures and invariants of links and 3-manifolds

本课题的研究属于三维拓扑学领域。这一领域的中心研究对象是称为3-流形的空间。三维流形是一种局部看起来像普通三维空间的对象，但其全局结构可能很复杂。三维拓扑学的一个重要部分也是对纽结(以某种纠缠方式嵌入到三维流形中的环)及其分类的研究。瑟斯顿的几何化猜想的解确立了3-流形(以及其中的纽结的补集)分解成允许显式几何的片断，并且双曲几何出现得更频繁。然而，在实践中，三维流形通常是以组合拓扑描述的形式给出的，从这些描述中寻找推导几何信息的方法是很自然和重要的。拓扑学家一直在研究三维流形的方法之一是通过使用不变量。在过去的几十年里，起源于物理学的想法导致了数学家们发现了各种纽结和三维流形的不变量。理解拓扑量、组合量和不变量与几何的关系是低维拓扑学的中心和重要目标。这个项目的主要主题是建立这种联系，并探索它们的分支和应用到其他数学领域。这个项目将建立几何和组合描述，性质，以及环和3-流形的量子不变量之间的关系。PI开发了一种设置，用于在有色Jones环多项式、环补中基本曲面的拓扑和几何以及双曲几何之间建立新的意想不到的关系。该项目的一部分将继续发展这一理论并探索其应用。另一部分，将结合几种新技术，开发从纯组合输入识别三维流形上的几何结构的方法，并从拓扑数据推导出几何量的估计。第三部分将研究三维流形中的斜链理论及其不变量，以及它与三维流形的几何分解的相互作用。第四部分将探索量子纽结不变量对纽结理论中经典问题的适用性，并寻找不改变潜在纽结拓扑的交叉变化的分类。该项目还涉及目前从事PI工作的研究生的研究。

The research of this project falls in the area of 3-dimensional topology. The central objects of study in this area are spaces called 3-manifolds. A 3-manifold is an object that locally looks like the ordinary 3- dimensional space but whose global structure can be complicated. An important part of 3-dimensional topology is also the study of knots (loops embedded in some tangled way in 3-manifolds) and their classification. The solution of Thurston's Geometrization Conjecture has established that 3-manifolds (and complements of knots in them) decompose into pieces that admit explicit geometries and that hyperbolic geometry is the one that appears more often. In practice, however, 3-manifolds are often given in terms of combinatorial topological descriptions and it is both natural and important to seek ways to deduce geometric information from these descriptions. One of the ways that topologists have been approaching the study of 3- manifolds is through the use of invariants. In the last few decades ideas originated in physics led mathematicians to the discovery of a variety of invariants of knots and 3-manifolds. Understanding the connections of topological and combinatorial quantities and invariants to geometry is a central and important goal of low dimensional topology. The main theme of this project is to establish such connections and explore their ramifications and applications to other areas of mathematics.This project will establish relationships between geometry and combinatorial descriptions, properties, and quantum invariants of links and 3-manifolds. The PI has developed a setting for establishing new unexpected relations between the colored Jones link polynomials, the topology and geometry of essential surfaces in link complements, and hyperbolic geometry. One part of the project will continue developing this theory and exploring its applications. Another part, will combine several new techniques, to develop methods for recognizing geometric structures on 3-manifolds from purely combinatorial input, and derive estimates on geometric quantities from topological data. A third part will study skein link theory in 3-manifolds, its invariants, and its interaction with geometric decompositions of 3-manifolds. A fourth part will explore the applicability of quantum knot invariants to classical questions in knot theory and search for a classification of crossing changes that do not alter the topology of the underlying knots. The project also involves the research of graduate students currently working with PI.